Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intregals:
$\int\frac{\sin2\text{x}}{(1+\sin\text{x})(2+\sin\text{x})}\text{ dx}$
✓

Answer

Let $\int\frac{\sin2\text{x}}{(1+\sin\text{x})(2+\sin\text{x})}\text{ dx}=\frac{\text{A}}{1+\sin\text{x}}+\frac{\text{B}}{2+\sin\text{x}}$
$\Rightarrow\sin2\text{x}=\text{A}(2+\sin\text{B})+\text{B}(1+\sin\text{B})$
$\Rightarrow2\sin\text{x}\cos\text{x}=(2\text{A}+\text{B})+(\text{A}+\text{B})\sin\text{x}$
Equating similar terms, we get,
$2\text{A}+\text{B}=0\Rightarrow\text{B}=-2\text{A}\text{ and}$
$\text{A}+\text{B}=2\cos\Rightarrow-\text{A}=2\cos\text{x}$
$\Rightarrow\text{A}=-2\cos\text{x}$
Thus,
$\text{I}=\int-\frac{2\cos\text{x}}{1+\sin\text{x}}\text{ dx}+\int\frac{4\cos\text{x}}{1+\sin\text{x}}\text{ dx}$
$=-2\log|1+\sin\text{x}|+4\log|2+\sin\text{x}|+\text{C}$
$\text{I}=\log\Big|\frac{(2+\sin\text{x})^4}{(1+\sin\text{x})^2}\Big|+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the intervals in which the following functions are increasing or decreasing.$f(x) = x^4- 4x$
→
If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi$ then show that $x^2+y^2+z^2+2 x y z=1$.
→
For what value of x the matrix A is singular?
$\text{A}=\begin{vmatrix}1+\text{x}&7\\3-\text{x}&8 \end{vmatrix}$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\log\text{x}$
→
D and E divide sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE.
→
Prove the following results:
$\tan^{-1}\frac{1}{7}+2\tan^{-1}\frac{1}{3}=\frac{\pi}{4}$
→
Prove that |A adj $A| = |A|^n.$
→
Find the area of the bounded by the curve $\text{y}=\frac{\pi}{2}+2\sin^{2}\text{x}$ x-axis and the area between x-axis, the curve and the ordinates $\text{x}=0, \text{x}=\pi$.
→
Evaluate the following integrals:
$\int(\text{x}+3)\sqrt{3-4\text{x}-\text{x}^2}\text{dx}$
→
Evaluate the following integrals:
$\int\big\{\tan(\log\text{x})+\sec^2(\log\text{x})\big\}\text{dx}$
→